By reviewing the primal-dual hybrid gradient algorithm (PDHG) proposed by He, You and Yuan (SIAM J. Image Sci., 7(4) (2014), pp. 2526--2537), in this paper we introduce four improved schemes for solving a class of saddle-point problems. Convergence properties of the proposed algorithms are ensured based on weak assumptions, where none of the objective functions are assumed to be strongly convex but the step-sizes in the primal-dual updates are more flexible than the previous. By making use of variational analysis, the global convergence and sublinear convergence rate in the ergodic/nonergodic sense are established, and the numerical efficiency of our algorithms is verified by testing an image deblurring problem compared with several existing algorithms.